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Mirrors > Home > ILE Home > Th. List > univ | GIF version |
Description: The union of the universe is the universe. Exercise 4.12(c) of [Mendelson] p. 235. (Contributed by NM, 14-Sep-2003.) |
Ref | Expression |
---|---|
univ | ⊢ ∪ V = V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwv 3705 | . . 3 ⊢ 𝒫 V = V | |
2 | 1 | unieqi 3716 | . 2 ⊢ ∪ 𝒫 V = ∪ V |
3 | unipw 4109 | . 2 ⊢ ∪ 𝒫 V = V | |
4 | 2, 3 | eqtr3i 2140 | 1 ⊢ ∪ V = V |
Colors of variables: wff set class |
Syntax hints: = wceq 1316 Vcvv 2660 𝒫 cpw 3480 ∪ cuni 3706 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-rex 2399 df-v 2662 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-uni 3707 |
This theorem is referenced by: (None) |
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